Question
Consider a function f(x) = x4 +
x3 + x2 + x + 1, where x is a positive integer greater than 1. What will be the remainder if f(x5) is divided by f(x) ?Solution
f(x) = x4 + x3 + x2 + x + 1 Multiply and divide by (x-1) f(x) = x4 + x3 + x2 + x + 1 X (x - 1) ÷ (x - 1) f(x) = x5 + x4 + x3 + x2 + x - x4 - x3 - x2 - x - 1 ÷ (x - 1) f(x) = (x5 −1) ÷ (x − 1) Now, f(x5) = x20 + x15 + x10 + x5 + 1 = x20 - 1 + 1 + x15 - 1 + 1 + x10 - 1 + 1 + x5 - 1 + 1 + 1 = (x20 - 1) + (x15 - 1) + (x10 - 1) + (x5 - 1) + 5 All the terms in the above is expression is divisible be (x5 - 1) except 5. Therefore, the remainder will be 5.
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